Classical Electromagnetism and Special Relativity - Covariant Formulation in Vacuum - 4-potential

4-potential

The EM field tensor can also be written

where

is the four-potential and

is the four-position.

Using the 4-potential in the Lorenz gauge, an alternative manifestly-covariant formulation can be found in a single equation (a generalization of an equation due to Bernhard Riemann by Arnold Sommerfeld, known as the Riemann–Sommerfeld equation, or the covariant form of the Maxwell equations):

Maxwell's equations (Covariant Lorenz gauge formulation)

where is the d'Alembertian operator, or four-Laplacian. For a more comprehensive presentation of these topics, see Covariant formulation of classical electromagnetism.